Question:
For a given number of states, what
is the largest number of 1's that can be printed by a beaver machine which halts?

Known busy beavers

Cases n = 1,2,3 solved by Lin and
Rado in 1960's

Case n = 4 solved by Brady in 1970's

Cases n = 5,6 had some monster
machines found in 1990's
and 2000's, but still open

bb(n):maximum number of
1's printed by a terminating beaver machine with n states (often
written Σ(n) )ff(n): maximum number of state
transitions made by a terminating beaver machine with n states (often
written S(n))prod(M): number of 1's printed by machine M
(undefined if M does not halt)

Hence bb(n) is
the maximum value of prod(M) for all
n-state machines M. Naturally ff(n)
≥ bb(n) and is typically much
larger. Possible that the values of bb(n) and ff(n) arise from
different machines.

For n =5 we "only' have to search through
230,400,000,000 = 2.3 ×1011 machines
...

Naive generate and test won't work; the trick is how to
incorporate the
test into the generation.

Input

1

2

3

4

5

B

O1 = 1, D1 = R,
N1 = 2

not (D3 = R, N3 =
2),
N3 ≠ h

O5, D5, N5

O7, D7, N7

O9, D9, N9

1

O2, D2, N2

O4, D4, N4

O6, D6, N6

O8, D8, N8

O10, D10, N10

Some "global" constraints can be used as well, such as avoiding N3
= 3,
D3 = R, N5 = 2, D5 = R.

Tree normal form:

emulate machine as the transitions are generated

add the halt state only when there is only one
"slot"
left

generate states in numerical order

detect loops as early as possible

Using equivalences and some other reduction techniques, it seems
the
number of 5-state candidates is "down" to 69,471,096 ie 6.9
×107
Loop detection is critical! Mining the data for the 117,440,512
machines with 4 states:

ones

5

6

7

8

9

10

11

12

13

machines

73,617

13,029

1981

475

79

13

6

5

2

89,207 machines terminate and print at least 5 1's

only 2,561 machines terminate and print more 1's
than
the 3-state busy beaver

loops abound!

Preliminary Results
for n=5 (or where is the white whale? :-)

Status

%

halts

31.9

cycle

5.2

"growing"

41.3

blank

21.5

unclassified

0.1

Note that classifying 99.9% of
the machines leaves "only" 0.1% or around 65,000 to classify ....

cycle means an identical tape configuration at two
different times

"growing" means reproducing the same
configuration shifted to the right or left)

An n-state machine which prints m 1's thus shows that bb(n)
≥ m and pp(m) ≤
n. pp(m):minimum number of
states of a terminating beaver machine which prints n 1'sww(m): minimum number of state
transitions made by a terminating beaver machine which prints n 1's

n

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

pp(n)

1

2

2

2

3

3

4

4

4

4

4

4

4

5

5

5

5

5

5

5

ww(n)

1

1

5

4

7

11

12

14

19

30

40

53

96

≤41

≤45

≤50

≤57

≤63

≤43

≤63

n

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

pp(n)

5

5

5

5

5

5

5

5

?

?

5

5

5

?

5

5

?

?

5

?

ww(n)

≤72

≤98

≤69

≤223

≤520

≤298

≤343

≤102

≤??

≤??

≤512

≤427

≤559

≤??

≤496

≤494

≤??

≤??

≤856

≤??

The table below gives the best 5-state machine to print the given
number of ones (which coincides with pp(n) for n >
14).

n

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

min hops

8

12

13

16

19

23

29

35

41

45

50

57

63

43

62

72

98

69

223

520

298

343

102

max hops

216

159

276

262

186

312

226

349

314

298

492

420

296

189

611

419

642

855

488

825

949

923

n

31

32

33

34

35

36

37

38

39

40

min hops

619

632

559

496

808

max hops

619

632

559

691

808

Not clear that we can fill in every entry in this table.Question: Is
it true that there is a 5-state machine which prints m 1's for each
bb(4) ≤ m ≤ bb(5)?

Note that this is certainly false for the range bb(5) to bb(6).
There are at least 10800 numbers to
cover, and "only" (4 × 7)12
= 232,218,265,089,212,416 = 2.3 × 10 17 machines.
So what is the distribution of placid platypus machines? Equivalence
(??)
Consider how one could print 6 1's.

Name

States

Ones

Hops

4

10

30

5

10

26

4

12

53

5

12

49

lb3

3

6

11

int4

4

6

9

int5

5

6

8

rr6

6

6

6

These are all equivalent in some sense. Can we generate the others
from
the 6-state version? Mysteries

Is bb(5) = 4098?

How to write a better loop detector

How to generate proofs of
non-termination from positive loop detection

Productivity for machines which are

contiguous (tape always of the form B1
*B)

eager (output is only 1, never B)

monotonic (no transitions with 1 input
and B output)

Distribution of 5-state machines
between bb(4) and bb(5)

Platypus numbers, ie those which can be
printed by a beaver machine. Known range at present is {1-24, 26-28,
32, 33, 1471,
4096, 4097, 4098, + a handful of large numbers}

What is the largest continuous range
which can be represented by a terminating beaver machine?

What is the smallest number that cannot
be printed by a 6-state (resp. 5-state) machine?

Relationship to 3n+1 problem

Constraints on tape access (cf. linear
bounded automata)

To Do List

Solve mysteries

Automatic drawing of machines

WYSIWYG editor for machines

Add features to programming suite
(stopping just short of a web browser :-)